Boil or steam all 3 pounds of potatoes until tender.
In a large pot, saute 1 chopped onion in olive oil until translucent, then add 1 quart of chicken broth, 1 cup water, and 3 cups of chopped broccoli florets.
Roughly chop all but one of the potatoes (set that last one aside for tomorrow) and add them to the broth.
Bring soup to a simmer until the broccoli is just cooked, then puree it with an immersion blender, leaving some chunks if you like.
Stir in 3 cups shredded cheddar cheese, salt and pepper to taste. Serve!